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## Risk

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**Prerequisites**Almost essential Consumption and Uncertainty Risk MICROECONOMICS Principles and Analysis Frank Cowell November 2006**Risk and uncertainty**• In dealing with uncertainty a lot can be done without introducing probability. • Now we introduce a specific probability model • This could be some kind of exogenous mechanism • Could just involve individual’s perceptions • Facilitates discussion of risk • Introduces new way of modelling preferences**Overview...**Risk Probability An explicit tool for model building Risk comparisons Special Cases Lotteries**Probability**• What type of probability model? • A number of reasonable versions: • Public observable • Public announced • Private objective • Private subjective • Need a way of appropriately representing probabilities in economic models. Lottery government policy? coin flip emerges from structure of preferences.**Ingredients of a probability model**• We need to define the support of the distribution • The smallest closed set whose complement has probability zero • Convenient way of specifying what is logically feasible (points in the support) and infeasible (other points). • Distribution function F • Represents probability in a convenient and general way. • Encompass both discrete and continuous distributions. • Discrete distributions can be represented as a vector • Continuous distribution – usually specify density function • Take some particular cases: a collection of examples**Some examples**• Begin with two cases of discrete distributions • #W = 2. Probability p of value x0; probability 1–p of value x1. • #W = 5. Probability pi of value xi, i = 0,...,4. • Then a simple example of continuous distribution with bounded support • The rectangular distribution – uniform density over an interval. • Finally an example of continuous distribution with unbounded support**Discrete distribution: Example 1**• Suppose of x0 and x1 are the only possible values. • Below x0 probability is 0. • Probability of x ≤ x0is p. 1 • Probability of x ≥ x0 but less than x1 is p. F(x) • Probability of x ≤ x1 is 1. p x x0 x1**Discrete distribution: Example 2**• There are five possible values: x0 ,…, x4. • Below x0 probability is 0. • Probability of x ≤ x0is p0. 1 • Probability of x ≤ x1 is p0+p1. • Probability of x ≤ x2 is p0+p1 +p2. p0+p1+p2+p3 • Probability of x ≤ x3 is p0+p1+p2+p3. F(x) p0+p1+p2 • Probability of x ≤ x4 is 1. p0+p1 • p0 + p1+ p2+ p3+ p4 = 1 p0 x x2 x3 x4 x0 x1**“Rectangular” : density function**• Suppose values are uniformly distributed between x0 and x1. • Below x0 probability is 0. f(x) x x0 x1**Rectangular distribution**• Values are uniformly distributed over the interval[x0,x1]. • Below x0 probability is 0. 1 • Probability of x ≥ x0 but less than x1 is [x x0] / [x1 x0] . • Probability of x ≤ x1 is 1. F(x) x x0 x1**Lognormal density**• Support is unbounded above. • The density function with parameters m=1, s=0.5 • The mean x 0 1 2 3 4 5 6 7 8 9 10**x**0 1 2 3 4 5 6 7 8 9 10 Lognormal distribution function 1**Overview...**Risk Probability Shape of the u-function and attitude to risk Risk comparisons Special Cases Lotteries**Risk aversion and the function u**• With a probability model it makes sense to discuss risk attitudes in terms of gambles • Can do this in terms of properties of “felicity” or “cardinal utility” function u • Scale and origin of uare irrelevant • But the curvature of uis important. • We can capture this in more than one way • We will investigate the standard approaches... • ...and then introduce two useful definitions**Risk aversion and choice**• Imagine a simple gamble • Two payoffs with known probabilities: • xREDwith probability pRED. • xBLUEwith probability pBLUE. • Expected value Ex = pREDxRED+ pBLUExBLUE. • A “fair gamble”: stake money is exactly Ex . • Would the person accept all fair gambles? • Compare Eu(x) with u(Ex) depends on shape of u**u**u(x) Risk-loving u u Risk-neutral Risk-averse u(x) u(x) x x xRED xRED xBLUE xBLUE Ex Ex x xRED xBLUE Ex Attitudes to risk • Shape of u associated with risk attitude • Neutrality: will just accept a fair gamble • Aversion: will reject some better-than-fair gambles • Loving: will accept some unfair gambles**pRED**– _____ pBLUE Risk premium and risk aversion • A given income prospect • The certainty equivalent income xBLUE • Slope gives probability ratio • Mean income • The risk premium. _ – • Risk premium: • Amount that amount you would sacrifice to eliminate the risk • Useful additional way of characterising risk attitude • P • P0 example xRED O x Ex**An example...**• Two-state model • Subjective probabilities (0.25, 0.75) • Single-commodity payoff in each case**Eu(x)** Ex Risk premium: an example u • Utility values of two payoffs • Expected payoff and the utility of expected payoff. • Expected utility and the certainty-equivalent u(Ex) u(x) u(xRED) • The risk premium again u(Ex) • Eu(x) amount you would sacrifice to eliminate the risk u(xBLUE) x xRED xBLUE x Ex x**Change the u-function**u • The utility function and risk premium as before • Now let the utility function become “flatter”… u(xRED) • Making the u-function less curved reduces the risk premium… • …and vice versa • More of this later u(xBLUE) u(xBLUE) x xRED x xBLUE x Ex**An index of risk aversion?**• Risk aversion associated with shape of u • second derivative • or “curvature” • But could we summarise it in a simple index or measure? • Then we could easily characterise one person as more/less risk averse than another • There is more than one way of doing this**Absolute risk aversion**• Definition of absolute risk aversion for scalar payoffs. uxx(x) a(x) := ux(x) • For risk-averse individuals a is positive. • For risk-neutral individuals a is zero. • Definition ensures that a is independent of the scale and the origin of u. • Multiply u by a positive constant… • …add any other constant… • a remains unchanged.**Relative risk aversion**• Definition of relative risk aversion for scalar payoffs: uxx(x) r(x) := x ux(x) • Some basic properties of r are similar to those of a: • positive for risk-averse individuals. • zero for risk-neutrality. • independent of the scale and the origin of u • Obvious relation with absolute risk aversion: • r(x) = xa(x)**Concavity and risk aversion**• Draw the function u again. u • Change preferences: φ is a concave function of u. • Risk aversion increases. lower risk aversion u(x) utility û(x) More concave uimplies higher risk aversion û =φ(u) higher risk aversion now to the interpretations x payoff**Interpreting a and r**• Think of a as a measure of the concavity of u. • Risk premium is approximately ½ a(x) var(x). • Likewise think of r as the elasticity of marginal u. • In both interpretations an increase in the “curvature” of u increases measured risk aversion. • Suppose risk preferences change… • u is replaced by û , where û=φ(u) and φ is strictly concave • Then both a(x) and r(x) increase for all x. • An increase in a or r also associated with increased curvature of IC…**Alf**xBLUE xBLUE Alf Charlie Bill xRED xRED O O Another look at indifference curves • Both uand p determine the shape of the IC • Alf and Bill differ in risk aversion • Alf and Charlie differ in subjective probability Same us but different ps Same ps but different us**Overview...**Risk Probability CARA and CRRA Risk comparisons Special Cases Lotteries**Special utility functions?**• Sometimes convenient to use special assumptions about risk. • Constant ARA • Constant RRA • By definition r(x) = xa(x) • Differentiate w.r.t. x: dr(x) da(x) = a(x) + x dx dx • So one could have, for example: • constant ARA and increasing RRA • constant RRA and decreasing ARA • or, of course, decreasing ARA and increasing RRA**Special case 1: CARA**• We take a special case of risk preferences • Assume that a(x) = a for all x • Felicity function must take the form 1 u(x) := eax a • Constant Absolute Risk Aversion • This induces a distinctive pattern of indifference curves...**Constant Absolute RA**• Case where a= ½ xBLUE • Slope of IC the same along 45° ray (standard vNM) • But, for CARA, slope of IC the same along any45° line xRED O**CARA: changing a**• Case where a = ½ (as before) xBLUE • Change ARA to a = 2 xRED O**Special case 2: CRRA**• Another important special case of risk preferences. • Assume that r(x) = r for all x. • Felicity function must take the form 1 u(x) := x1 r 1 r • Constant Relative Risk Aversion • Again induces a distinctive pattern of indifference curves...**Constant Relative RA**• Case where r= 2 xBLUE • Slope of IC the same along 45° ray (standard vNM) • For CRRA, slope of IC is the same along any ray through O • ICs are homothetic. xRED O**CRRA: changing r**• Case where r= 2 (as before) xBLUE • Change RRA to r= ½ xRED O**CRRA: changing p**• Case where r= 2 (as before) xBLUE • Increase probability of state RED xRED O**Overview...**Risk Probability Probability distributions as objects of choice Risk comparisons Special Cases Lotteries**Lotteries**• Consider lottery as a particular type of uncertain prospect. • Take an explicit probability model. • Assume a finite number of states-of-the-world • Associated with each state w are: • A known payoff xw , • A known probabilitypw ≥ 0. • The lottery is the probability distribution over the “prizes” xw, w=1,2,...,. • The probability distribution is just the vector p:= (p1,,p2 ,…,,p) • Of course, p1+ p2 +…+p = 1. • What of preferences?**The probability diagram: #W=2**• Probability of state RED • Probability of stateBLUE pBLUE • Cases of perfect certainty. • Cases where 0 < p< 1 • The case (0.75, 0.25) • (0,1) pRED+pBLUE= 1 • Only points on the purple line make sense. • This is an 1-dimensional example of a simplex • (0, 0.25) (0.75, 0) pRED • (1,0)**The probability diagram: #W=3**pBLUE • Third axis corresponds to probability of state GREEN • There are now three cases of perfect certainty. • (0,0,1) • Cases where 0 < p< 1 • The case (0.5, 0.25, 0.25) pRED + pGREEN + pBLUE= 1 pGREEN • Only points on the purple triangle make sense, • This is a 2-dimensional example of a simplex • • (0,1,0) (0, 0, 0.25) (0, 0.25 , 0) 0 • (1,0,0) (0.5, 0, 0) pRED**Probability diagram #W=3 (contd.)**• All the essential information is in the simplex • (0,0,1) • Display as a plane diagram • The equi-probable case • The case (0.5, 0.25, 0.25) •(1/3,1/3,1/3) •(0.5, 0.25, 0.25) • (0,1,0) • . (1,0,0)**Preferences over lotteries**• Take the probability distributions as objects of choice • Imagine a set of lotteries p°, p', p",... • Each lottery p has same payoff structure • State-of-the-world w has payoff xw • ... and probabilitypw° or pw' or pw" ... depending on which lottery • We need an alternative axiomatisation for choice amongst lotteries p**Axioms on preferences**• Transitivity over lotteries • If p°<p' and p'<p" ... • ...then p°<p". • Independence of lotteries • If p°<p' and l(0,1)... • ...then lp° + [1-l]p" <lp' + [1-l]p" • Continuity over lotteries • If p°Âp'Âp" then there are numbers l and m such that • lp° +[1-l]p" Â p' • p' Â mp° +[1-m]p"**Basic result**• Take the axioms transitivity, independence, continuity • Imply that preferences must be representable in the form of a von Neumann-Morgenstern utility function: åpw u(xw) w ÎW • or equivalently: åpw uw w ÎW whereuw := u(xw) • So we can also see the EU model as a weighted sum of ps.**p-indifference curves**• Indifference curves over probabilities. • (0,0,1) • Effect of an increase in the size of uBLUE • (0,1,0) • . (1,0,0)**What next?**• Simple trading model under uncertainty • Consumer choice under uncertainty • Models of asset holding • Models of insurance • This is in the presentation Risk Taking